Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
270 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,14)(8,9)(10,13)(11,15)(12,18)(16,17);(6,9)(7,15)(8,18)(11,17)(12,13)(14,19)(16,20);(2,4)(5,7,8,16,10,17,18,14)(6,11,12,20,13,15,9,19)Polycyclic presentation:
2^8|c^2=d,d^2=h,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=dh,e^b=eg,f^b=fg,g^b=gh,e^c=egh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125721, #125722, #125733, … |
| maximal normal subgroup of | #125721, #125722, #125733, … |
| maximal quotient of | #144892, #144925, #145011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6401);